holt algebra 2 12-5 mathematical induction and infinite geometric series 12-5 mathematical induction...

Holt Algebra 2

12-5 Mathematical Induction and Infinite Geometric Series12-5 Mathematical Induction and

Infinite Geometric Series

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

12-5 Mathematical Induction and Infinite Geometric Series

Warm UpEvaluate.

1. 2.

3. Write 0.6 as a fraction in simplest form.

10

4. Find the indicated sum for the geometric

series 3

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Find sums of infinite geometric series.

Use mathematical induction to prove statements.

Objectives

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infinite geometric seriesconvergelimitdivergemathematical induction

Vocabulary

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In Lesson 12-4, you found partial sums of geometric series. You can also find the sums of some infinite geometric series. An infinite geometric series has infinitely many terms. Consider the two infinite geometric series below.

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Notice that the series Sn has a common ratio of and the partial sums get closer and closer to 1 as n increases. When |r|< 1 and the partial sum approaches a fixed number, the series is said to converge. The number that the partial sums approach, as n increases, is called a limit.

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For the series Rn, the opposite applies. Its common ratio is 2, and its partial sums increase toward infinity. When |r| ≥ 1 and the partial sum does not approach a fixed number, the series is said to diverge.

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Example 1: Finding Convergent or Divergent Series

Determine whether each geometric series converges or diverges.

A. 10 + 1 + 0.1 + 0.01 + ... B. 4 + 12 + 36 + 108 + ...

The series converges and has a sum.

The series diverges anddoes not have a sum.

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Check It Out! Example 1

Determine whether each geometric series converges or diverges.

A. B. 32 + 16 + 8 + 4 + 2 + …

The series converges and has a sum.

The series diverges anddoes not have a sum.

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If an infinite series converges, we can find the sum.

Consider the series

from the previous page. Use the formula for the

partial sum of a geometric series with and

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Graph the simplified equation on a graphing calculator. Notice that the sum levels out and converges to 1.

As n approaches infinity, the term approaches zero. Therefore, the sum of the series is 1. This concept can be generalized for all convergent geometric series and proved by using calculus.

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Find the sum of the infinite geometric series, if it exists.

Example 2A: Find the Sums of Infinite Geometric Series

1 – 0.2 + 0.04 – 0.008 + ...

r = –0.2 Converges: |r| < 1.

Sum formula

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Example 2A Continued

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Evaluate.

Converges: |r| < 1.

Example 2B: Find the Sums of Infinite Geometric Series


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Example 2B Continued

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Check It Out! Example 2a


r = –0.2 Converges: |r| < 1.

Sum formula

125

6=

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Check It Out! Example 2b Find the sum of the infinite geometric series, if it exists.

Evaluate.

Converges: |r| < 1

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You can use infinite series to write a repeating decimal as a fraction.

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Example 3: Writing Repeating Decimals as Fractions

Write 0.63 as a fraction in simplest form.

Step 1 Write the repeating decimal as an infinite geometric series.

0.636363... = 0.63 + 0.0063 + 0.000063 + ...

Use the pattern for the series.

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Example 3 Continued

Step 2 Find the common ratio.

|r | < 1; the series converges to a sum.

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Example 3 Continued

Step 3 Find the sum.

Apply the sum formula.

Check Use a calculator to divide the fraction

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Recall that every repeating decimal, such as 0.232323..., or 0.23, is a rational number and can be written as a fraction.

Remember!

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Write 0.111… as a fraction in simplest form.

Step 1 Write the repeating decimal as an infinite geometric series.

0.111... = 0.1 + 0.01 + 0.001 + ...Use the pattern for the series.

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Step 2 Find the common ratio.

|r | < 1; the series converges to a sum.

Check It Out! Example 3 Continued

Step 3 Find the sum.

Apply the sum formula.

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You have used series to find the sums of many sets of numbers, such as the first 100 natural numbers. The formulas that you used for such sums can be proved by using a type of mathematical proof called mathematical induction.

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Use mathematical induction to prove that

Example 4: Proving with Mathematical Induction

Step 1 Base case: Show that the statement is true for n = 1.

The base case is true.

Step 2 Assume that the statement is true for a natural number k.

Replace n with k.

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Example 4 Continued

Step 3 Prove that it is true for the natural number k + 1.

Find the common denominator.

Add the next term (k + 1) to each side.

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Example 4 Continued

Write with (k + 1).

Factor out (k + 1).

Simplify.

Add numerators.

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Use mathematical induction to prove that the sum of the first n odd numbers is

1 + 3 + 5 + … +(2n - 1) = n2.

Step 1 Base case: Show that the statement is true for n = 1.

(2n – 1) = n2

2(1) – 1 = 12

1 = 12

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Check It Out! Example 4 Continued

Step 2 Assume that the statement is true for a natural number k.

1 + 3 + … (2k – 1) = k2

1 + 3 + … (2k – 1) + [2(k + 1) – 1]

Step 3 Prove that it is true for the natural number k + 1.

= k2 + [2(k + 1) – 1]

= (k + 1)2

= k2 + 2k + 1

Therefore, 1 + 3 + 5 + … + (2n – 1) = n2.

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Identify a counterexample to disprove a3 > a2, where a is a whole number.

Example 5: Using Counterexamples

33 > 32 23 > 22 13 > 12 03 > 02

27 > 9 8 > 4 1 > 1 0 > 0

a3 > a2 is not true for a = 1, so it is not true for all whole numbers. 0 is another possible counterexample.

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Often counterexamples can be found using special numbers like 1, 0, negative numbers, or fractions.

Helpful Hint

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Identify a counterexample to disprove

where a is a real number.

is not true for a = 5, so it is not true

for all real numbers.

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Lesson Quiz

Solve each equation.

1. Determine whether the geometric series150 + 30 + 6 + … converges or diverges, and find the sum if it exists. converges; 187.5

4909

3. Either prove by induction or provide a counterexample to disprove the following statement: 1 + 2 + 3 + 4 + … +

counterexample: n = 2

2. Write 0.0044 as a fraction in simplest form.

holt algebra 2 12-5 mathematical induction and infinite geometric series 12-5 mathematical induction...

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